Blow-up solutions of the "bad" Boussinesq equation

Abstract

We study blow-up solutions of the ``bad" Boussinesq equation, and prove that a wide range of asymptotic scenarios can happen. For example, for each $T>0$, $x_0\in \mathbb{R}$ and $\delta \in (0,1)$, we prove that there exist Schwartz class solutions $u(x,t)$ on $\mathbb{R}\times [0,T)$ such that $|u(x,t)|\le C\frac{1+x^2}{(x-x_0{)}^2}$ and $u(x_0,t)\asymp (T-t{)}^{-\delta }$ as $t\to T$. We also prove that for any $q\in \mathbb{N}$, $T>0$, $x_0\in \mathbb{R}$, $\delta \in (0,\frac{1}{2})$, there exist Schwartz class solutions $u(x,t)$ on $\mathbb{R}\times [0,T)$ such that (i) $|{\partial }_x^{q_1}{\partial }_t^{q_2}u(x,t)|\le C$ for each $q_1,q_2\in \mathbb{N}$ such that $q_1+2q_2\le q$, (ii) $|{\partial }_x^{q_1}{\partial }_t^{q_2}u(x,t)|\le C\frac{1+|x|}{|x-x_0|}$ for each $q_1,q_2\in \mathbb{N}$ such that $q_1+2q_2=q+1$, (iii) $|{\partial }_x^{q_1}{\partial }_t^{q_2}u(x_0,t)|\asymp (T-t{)}^{-\delta }$ as $t\to T$ for each $q_1,q_2\in \mathbb{N}$ such that $q_1+2q_2=q+1$. In particular, when $q=0$, this result establishes the existence of wave-breaking solutions, i.e. solutions that remain bounded but whose $x$-derivative blows up in finite time.

32 pages, 2 figures